Cofactor (mathematics)

In mathematics, a cofactor is a component of a matrix computation of the matrix determinant.

Let M be a square matrix of size n. The (i,j) minor refers to the determinant of the (n-1)×(n-1) submatrix M_i,j formed by deleting the i-th row and j-th column from M (or sometimes just to the submatrix M_i,j itself). The corresponding cofactor is the signed determinant

$(-1)^{i+j}\det M_{i,j}.\,$

The adjugate matrix adj M is the square matrix whose (i,j) entry is the (j,i) cofactor. We have

$M\cdot \mathop {\mbox{adj}} M=(\det M)I_{n}=\mathop {\mbox{adj}} M\cdot M,\,$

which encodes the rule for expansion of the determinant of M by any the cofactors of any row or column. This expression shows that if det M is invertible, then M is invertible and the matrix inverse is determined as

$M^{-1}=(\det M)^{-1}\mathop {\mbox{adj}} M.\,$

Example

Consider the following example matrix,

$M={\begin{pmatrix}a_{1}&a_{2}&a_{3}\\b_{1}&b_{2}&b_{3}\\c_{1}&c_{2}&c_{3}\\\end{pmatrix}}.$

Its minors are the determinants (bars indicate a determinant):

$M_{11}={\begin{vmatrix}b_{2}&b_{3}\\c_{2}&c_{3}\\\end{vmatrix}}\quad M_{12}={\begin{vmatrix}b_{1}&b_{3}\\c_{1}&c_{3}\\\end{vmatrix}}\quad M_{13}={\begin{vmatrix}b_{1}&b_{2}\\c_{1}&c_{2}\\\end{vmatrix}}\quad M_{21}={\begin{vmatrix}a_{2}&a_{3}\\c_{2}&c_{3}\\\end{vmatrix}}\quad M_{22}={\begin{vmatrix}a_{1}&a_{3}\\c_{1}&c_{3}\\\end{vmatrix}}\quad$

$M_{23}={\begin{vmatrix}a_{1}&a_{2}\\c_{1}&c_{2}\\\end{vmatrix}}\quad M_{31}={\begin{vmatrix}a_{2}&a_{3}\\b_{2}&b_{3}\\\end{vmatrix}}\quad M_{32}={\begin{vmatrix}a_{1}&a_{3}\\b_{1}&b_{3}\\\end{vmatrix}}\quad M_{33}={\begin{vmatrix}a_{1}&a_{2}\\b_{1}&b_{2}\\\end{vmatrix}}\quad$

The adjugate matrix of M is

$A={\begin{pmatrix}M_{11}&-M_{21}&M_{31}\\-M_{12}&M_{22}&-M_{32}\\M_{13}&-M_{23}&M_{33}\\\end{pmatrix}},$

and the inverse matrix is

$M^{-1}=|M|^{-1}A\,.$

Indeed,

${\begin{aligned}\left(M\;M^{-1}\right)_{11}&=|M|^{-1}\left(a_{1}M_{11}-a_{2}M_{12}+a_{3}M_{13}\right)={\frac {|M|}{|M|}}=1\\\left(M\;M^{-1}\right)_{21}&=|M|^{-1}\left(b_{1}M_{11}-b_{2}M_{12}+b_{3}M_{13}\right)=b_{1}(b_{2}c_{3}-b_{3}c_{2})-b_{2}(b_{1}c_{3}-b_{3}c_{1})+b_{3}(b_{1}c_{2}-b_{2}c_{1})=0,\\\end{aligned}}$

and the other matrix elements of the product follow likewise.

References

• C.W. Norman (1986). Undergraduate Algebra: A first course. Oxford University Press, 306,310,315. ISBN 0-19-853248-2.